Elliptic Two-Dimensional Invariant Tori for the Planetary Three-Body Problem

The spatial planetary three-body problem (i.e., one star and two planets, modelled by three massive points, interacting through gravity in a three dimensional space) is considered. It is proved that, near the limiting stable solutions given by the two planets revolving around the star on Keplerian ellipses with small eccentricity and small non-zero mutual inclination, the system affords two-dimensional, elliptic, quasi-periodic solutions, provided the masses of the planets are small enough compared to the mass of the star and provided the osculating Keplerian major semi-axes belong to a two-dimensional set of density close to one.     

A visualization study of the interaction of a free vortex with the wake behind an airfoil

The two-dimensional interaction of a single vortex with a thin symmetrical airfoil and its vortex wake has been investigated in a low turbulence wind tunnel having velocity of about 2 m/s in the measuring section. The flow Reynolds number based on the airfoil chord length was 4.5 × 10³. The investigation was carried out using a smoke-wire visualization technique with some support of standard hot-wire measurements. The experiment has proved that under certain conditions the vortex-airfoil-wake interaction leads to the formation of new vortices from the part of the wake positioned closely to the vortex. After the formation, the vortices rotate in the direction opposite to that of the incident vortex.List of symbols c test airfoil chord - C vortex generator airfoil chord - TA test airfoil - TE test airfoil trailing edge - TE _G vortex generator airfoil trailing edge - t dimensionless time-interval measured from the vortex passage by the test airfoil trailing edge: gDt=(T-T- _TE)·U/c - T time-interval measured from the start of VGA rotation - U free stream velocity - U vortex induced velocity fluctuation - VGA vortex generator airfoil - y distance in which the vortex passes the test airfoil - Z vortex circulation coefficient: Z=/(U · c/2) - vortex generator airfoil inclination angle - vortex circulation - vortex strength: =/2     

Recording and evaluation methods of PIV investigations on a helicopter rotor model

Three-component particle image velocimetry measurements at moderate speeds and observation distances can now be accomplished on a routine basis. This article discusses the experiment performed on a 4 m-diameter model rotor in the 6-m×8-m open test section of the Large Low Speed Facility of the German–Dutch Wind Tunnels. More than half a terabyte of raw data were recorded at various positions on the advancing and retreating sides of the rotor in order to obtain detailed measurements of the trailing vortex in the frame of an international project. This paper addresses measuring techniques and possible sources of errors and presents a limited number of cases for the purpose of illustrating the solutions to numerous technical challenges relating to the acquisition and analysis of vortical flows.List of symbols C _T thrust coefficient (T/² R ⁴) - M magnification - r _c radius of vortex core (mm) - R rotor radius (m) - T thrust (N) - u,v,w velocity components in x, y and z coordinates (m/s) - (u,v,w)_wt velocity components in wind tunnel coordinates (m/s) - U _max maximum in-plane velocity component (m/s) - W _max maximum out-of-plane velocity component (m/s) - x,y,z particle image velocimetry (PIV) frame coordinates (m) - (x,y,z)_wt wind tunnel coordinates (m) - t time delay (s) - Z light sheet thickness (mm) - Z light sheet thickness (mm) - rotor rotation frequency (rad/s) - rotor azimuth angle during recording (deg) vortex age - rotor shaft angle (deg) - _x displacement measurement error - advance ratio (V/R) - air density (kg/m³) - circulation (m²/s) - _z vorticity (s^–1) Abbreviations AFDD Aeroflightdynamics Directorate - BVI blade–vortex interaction - DLR Deutches zentrum für Luft- und Raumfahrt - DNW German–Dutch Wind tunnel - HART HHC aeroacoustic rotor test - LLF large low speed facility - NASA National Aeronautics and Space Administration - ONERA Office National dEtudes e de Recherches Aerospatiales - RANS Reynolds-averaged Navier–Stokes - SPR stereo pattern recognition - 3C-PIV three-component particle image velocimetry     

Theoretical derivation of tangential velocity profiles in a flat vortex chamber-influence of turbulence and wall friction

Summary As part of a study on the hydrodynamics of a cyclone separator, a theoretical investigation of the flow pattern in a flat box cyclone (vortex chamber) has been carried out. Expressions have been derived for the tangential velocity profile as influenced by internal friction (eddy viscosity) and wall friction. The most important parameter controlling the tangential velocity profile is = –u ₀ R/(v+ ), where u ₀ is the radial velocity at the outer radius R of the cyclone, the kinematic liquid viscosity and is the kinematic eddy viscosity. For values of greater than about 10 the tangential velocity profile is nearly hyperbolic, for smaller than 1 the tangential velocity even decreases towards the centre. It is shown how and also the wall friction coefficient may be obtained from experimental velocity profiles with the aid of suitable graphs. Because of the close relation between eddy viscosity and eddy diffusion, measurements of velocity profiles in flat box cyclones will also provide information on the eddy motion of particles in a cyclone, a motion reducing its separation efficiency.List of symbols A cross-sectional area of cyclone inlet - h height of cyclone - p static pressure in cyclone - p static pressure difference in cyclone between two points on different radius - r radius in cyclone - r ₁ radius of cyclone outlet - R radius of cyclone circumference - u radial velocity in cyclone - u ₀ radial velocity at circumference of flat box cyclone - v tangential velocity - v ₀ tangential velocity at circumference of flat box cyclone - w axial velocity - z axial co-ordinate in cyclone - friction coefficient in flat box cyclone (for definition see § 5) - ₁ value of friction coefficient for ₁<< ₂ - ₂ value of friction coefficient for ₂<<1 - = - ₁ value of for ₁<< ₂ - ₂ value of for ₂<<1 - thickness of laminar boundary layer - =/h - turbulent kinematic viscosity - ratio of z to h - k ratio of height of cyclone to radius R of cyclone - parameter describing velocity profile in cyclone =–u ₀ R/(+) - kinematic viscosity of fluid - density of fluid - ratio of r to R - ₁ value of at outlet of cyclone - ₂ value of at inner radius of cyclone inlet - _w shear stress at cyclone wall - angular momentum in cyclone/angular momentum in cyclone inlet - ₁ value of at = ₁ - ₂ value of at = ₂     

Designing multi-link function mechanisms by passing conventional boundaries

The mechanism synthesis concerning pure numerical methods can only be realized below given boundaries — and this because of the high power of the system equations. For some time the author has been using graphical methods to pass those boundaries concerning multilink mechanisms. By means of the Drawing-Following Method developed by the author, simple but highly accurate computer programs are now available. As a fundamental rule, reductions are used by the coincidences of several poles. Additionally, the new methods are based on the first deviation of correlation of point positions and relative angles, i.e. by the use of the momentary velocity ratios. By this the torque and force relationships are also immediately given.This paper is a renewed part of a series of lectures given by the author between 25 and 28 September 1972 at the International Centre for Mechanical Sciences, Udine, Italy. The lectures were entitled Theory of mechanisms, dimensional synthesis of linkages by geometrical and graphical methods (10 lecture hours). In the meantime those lectures have been computer programmed using the so-called Drawing-Following Method developed by the author.     

Qualitative behavior of dissipative wave equations on bounded domains

The qualitative behavior of solutions of the mixed problem u_tt = u-a(x)u_t in IR x , u=0 on IR x , is studied in the case when a>0 and IRⁿ is bounded. Roughly speaking, if aa_min>0, then solutions decay at least as fast as exp t( –1/2a_min), with the possible exception of a finite dimensional set of smooth solutions whose existence is associated with a phenomenon of overdamping. If a_max is sufficiently small, depending on , then no overdamping occurs.Partially supported by NSF grant NSF GP 34260.This work was partially supported by the National Science Foundation under Grant No. GP 34260     

The dynamic performance of the Weissenberg rheogoniometer III. Large amplitude oscillatory shearing — harmonic analysis

The harmonic content of the nonlinear dynamic behaviour of 1% polyacrylamide in 50% glycerol/water was studied using a standard Model R 18 Weissenberg Rheogoniometer. The Fourier analysis of the Oscillation Input and Torsion Head motions was performed using a Digital Transfer Function Analyser.In the absence of fluid inertia effects and when the amplitude of the (fundamental) Oscillation Input motion _I is much greater than the amplitudes of the Fourier components of the Torsion Head motion _Tn empirical nonlinear dynamic rheological propertiesG _n (, ⁰),G _n (, ⁰) and/or _n (, ⁰), _n (, ⁰) may be evaluated without a-priori-knowledge of a rheological constitutive equation. A detailed derivation of the basic equations involved is presented.Cone and plate data for the third harmonic storage modulus (dynamic rigidity)G ₃ (, ⁰), loss modulusG ₃ (, ⁰) and loss angle ₃ (, ⁰) are presented for the frequency range 3.14 × 10^–2 1.25 × 10² rad/s at two strain amplitudes, _CP ⁰ = 2.27 and 4.03. Composite cone and plate and parallel plates data for both the third and fifth harmonic dynamic viscosities ₃ (, ⁰), _S (, ⁰) and dynamic rigiditiesG ₃ (, ⁰),G ₅ (, ⁰) are presented for strain amplitudes in the ranges 1.10 _CP ⁰ 4.03 and 1.80 _PP ⁰ 36 for a single frequency, = 3.14 × 10^–1 rad/s. Good agreement was obtained between the results from both geometries and the absence of significant fluid inertia effects was confirmed by the superposition of the data for different gap widths.     

Optical co-ordinates in general relativity 2 — the problem of distance

Summary Let denote the congruence of null geodesics associated with a given optical observer inV ₄. We prove that determines a unique collection of vector fieldsM₍₎ ( =1, 2, 3) and ₍₀₎ overV ₄, satisfying a weak version of Killing's conditions.This allows a natural interpretation of these fields as the infinitesimal generators of spatial rotations and temporal translation relative to the given observer. We prove also that the definition of the fieldsM₍₎ and ₍₀₎ is mathematically equivalent to the choice of a distinguished affine parameter f along the curves of, playing the role of a retarded distance from the observer.The relation between f and other possible definitions of distance is discussed.

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Sommario Sia la congruenza di geodetiche nulle associata ad un osservatore ottico assegnato nello spazio-tempoV ₄. Dimostriamo che determina un'unica collezione di campi vettorialiM₍₎ ( =1, 2, 3) e ₍₀₎ inV ₄ che soddisfano una versione in forma debole delle equazioni di Killing. Ciò suggerisce una naturale interpretazione di questi campi come generatori infinitesimi di rotazioni spaziali e traslazioni temporali relative all'osservatore assegnato. Dimostriamo anche che la definizione dei campiM₍₎, ₍₀₎ è matematicamente equivalente alla scelta di un parametro affine privilegiato f lungo le curve di, che gioca il ruolo di distanza ritardata dall'osservatore. Successivamente si esaminano i legami tra f ed altre possibili definizioni di distanza in grande.

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Work performed in the sphere of activity of: Gruppo Nazionale per la Fisica Matematica del CNR.     

About the Regularized Navier–Stokes Equations

The first goal of this paper is to study the large time behavior of solutions to the Cauchy problem for the 3-dimensional incompressible Navier–Stokes system. The Marcinkiewicz space L^3, is used to prove some asymptotic stability results for solutions with infinite energy. Next, this approach is applied to the analysis of two classical regularized Navier–Stokes systems. The first one was introduced by J. Leray and consists in mollifying the nonlinearity. The second one was proposed by J.-L. Lions, who added the artificial hyper-viscosity (–)^{/ 2}, > 2 to the model. It is shown in the present paper that, in the whole space, solutions to those modified models converge as t toward solutions of the original Navier–Stokes system.     

Invariant Manifolds for Nonautonomous Systems with Application to One-Step Methods

In this paper we study the existence of invariant manifolds for a special type of nonautonomous systems which arise in the study of discretization methods. According to [10], a one-step scheme of step-size for an autonomous system can be interpreted as the -flow of a perturbed nonautonomous system. The perturbation is rapidly forced in the sense that it is periodic with respect to time with period . Assuming a saddle node for the autonomous system, we prove that these rapidly forced perturbations have center manifolds which exist in a uniform neighborhood and which converge to a center manifold of the autonomous system as tends to zero. Our results are applied to obtain a smooth continuation as well as estimates of the well known center manifolds for one-step schemes. They also form the basis for studying saddle-node homoclinic orbits under discretization.     

Über singuläre Lastfälle in einer linearen Schalentheorie und ihre finite Behandlung

Übersicht Ausgehend von bekannten Fundamentallösungen für Platten bzw. Scheiben wird die Erstellung singulärer Ansatzfunktionen gezeigt, wie sie für finite Näherungsverfahren benötigt werden, die von den Funktionalen der totalen Energie bzw. der komplementären Energie ausgehen. Das Vorgehen wird eingehend an Kreiszylinderschalen erläutert.

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Summary Starting from known fundamental solutions of plates the construction of singular basic estimate functions is shown. These are necessary in finite approximation methods basing on the functionals of total energy or complementary energy. The proceeding is explained in detail in a cylindrical shell analysis.

     

Parameter determination for thermodynamical models of the pseudoelastic behaviour of shape memory alloy by resistivity measurements and infrared thermography

Two thermodynamical models of pseudoelastic behaviour of shape memory alloys have been formulated. The first corresponds to the ideal reversible case. The second takes into account the hysteresis loop characteristic of this shape memory alloys.Two totally independent techniques are used during a loading-unloading tensile test to determine the whole set of model parameters, namely resistivity and infrared thermography measurements. In the ideal case, there is no difficulty in identifying parameters.Infrared thermography measurements are well adapted for observing the phase transformation thermal effects.Notations 1 austenite 2 martensite - ₍₎ Macroscopic infinitesimal strain tensor of phase - ₍₂₎ ^f Traceless strain tensor associated with the formation of martensite phase - Macroscopic infiniesimal strain tensor - Macroscopic infinitesimal strain tensor deviator - _f Trace - Equivalent strain - ^pe Macroscopic pseudoelastic strain tensor - x Distortion due to parent (austenite =1)product (martensite =2) phase transformation (traceless symmetric second order tensor) - M Total mass of a system - M() Total mass of phase - V Total volume of a system - V() Total volume of phase - z=M(2)/M Weight fraction of martensite - 1-z=M(1)/M Weight fraction of austenite - u ₀ ^* () Specific internal energy of phase (=1,2) - s ₀ ^* () Specific internal entropy of phase - Specific configurational energy - Specific configurational entropy - ₀ ^f (T) Driving force for temperature-induced martensitic transformation at stress free state ( ₀ ^f T) = T ^*–Ts ^*) - Kirchhoff stress tensor - Kirchhoff stress tensor deviator - Equivalent stress - Cauchy stress tensor - Mass density - K Bulk moduli (K ₀=K) - L Elastic moduli tensor (order 4) - E Young modulus - Energetic shear (₀ = ) - Poisson coefficient - M _s ^o (M _F ^o ) Martensite start (finish) temperature at stress free state - A _s ^o (A _F ^o ) Austenite start (finish) temperature at stress free state - C _v Specific heat at constant volume - k Conductivity - Pseudoelastic strain obtained in tensile test after complete phase transformation (AM) (unidimensional test) - ₀ Thermal expansion tensor - r Resistivity - 1MPa 10⁶ N/m ² - ⁽⁾ Specific free energy of phase - _n Specific free energy at non equilibrium (R model) - _n ^eq Specific free energy at equilibrium (R model) - _n ^v Volumic part of ^eq - Specific free energy at non equilibrium (R _L model) - _conf Specific coherency energy (R _L model) - _c Specific free energy at constrained equilibria (R _L model) - _it (T) Coherency term (R _L model)     

Eine analytische Näherungslösung für die eingefrorene laminare Grenzschichtströmung eines binären Gemisches

Zusammenfassung Für die eingefrorene laminare Grenzschichtströmung eines teilweise dissoziierten binären Gemisches entlang einer stark gekühlten ebenen Platte wird eine analytische Näherungslösung angegeben. Danach läßt sich die Wandkonzentration als universelle Funktion der Damköhler-Zahl der Oberflächenreaktion angeben. Für das analytisch darstellbare Konzentrationsprofil stellt die Damköhler-Zahl den Formparameter dar. Die Wärmestromdichte an der Wand bestehend aus einem Wärmeleitungs- und einem Diffusionsanteil wird angegeben und diskutiert. Das Verhältnis beider Anteile läßt sich bei gegebenen Randbedingungen als Funktion der Damköhler-Zahl ausdrücken.

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An analytical approximation for the frozen laminar boundary layer flow of a binary mixture

An analytical approximation is derived for the frozen laminar boundary layer flow of a partially dissociated binary mixture along a strongly cooled flat plate. The concentration at the wall is shown to be a universal function of the Damkohler-number for the wall reaction. The Damkohlernumber also serves as a parameter of shape for the concentration profile which is presented in analytical form. The heat transfer at the wall depending on a conduction and a diffusion flux is derived and discussed. The ratio of these fluxes is expressed as a function of the Damkohler-number if the boundary conditions are known.

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Formelzeichen A Atom - A₂ Molekül - C Konstante in Gl. (20) - c₁=1/(2C) Konstante in Gl. (35) - c_p spezifische Wärme bei konstantem Druck - D binärer Diffusionskoeffizient - Ec=u ² /(2h_f) Eckert-Zahl - h spezifische Enthalpie - h_t=h+u²/2 totale spezifische Enthalpie - h _A ⁰ spezifische Dissoziationsenthalpie - K_w Reaktionsgeschwindigkeitskonstante der heterogenen Wandreaktion - 1= /( ) Champman-Rubesin-Parameter - Le=Pr/Sc Lewis-Zahl - M Molmasse - p statischer Druck - Pr= c_pf/ Prandtl-Zahl - q_w Wärmestromdichte an der Wand - q_cw, q_dw Wärmeleitungsbzw. Diffusionsanteil der Wärmestromdichte an der Wand - universelle Gaskonstante - R=/(2M_a) individuelle Gaskonstante der molekularen Komponente - Re_x= u x/ Reynolds-Zahl - Sc=/( D) Schmidt-Zahl - T absolute Temperatur - T_d=h _A ⁰ /R charakteristische Dissoziationstemperatur - u, v x- und y-Komponenten der Geschwindigkeit - U=u/u normierte x-Komponente der Geschwindigkeit - x, y Koordinaten parallel und senkrecht zur Platte Griechische Symbole - =A/ Dissoziationsgrad - Grenzschichtdicke - ₂ Impulsverlustdicke - Damköhler-Zahl der Oberflächenreaktion - =T/T normierte Temperatur - =y/ normierter Wandabstand - Wärmeleitfähigkeit - dynamische Viskosität - , ^* Ähnlichkeitskoordinaten - Dichte - Schubspannung Indizes A auf ein Atom bezogen - M auf ein Molekül bezogen - f auf den eingefrorenen Zustand bezogen - w auf die Wand bezogen - auf den Außenrand der Grenzschicht bezogen     

Representation of the rheological properties of polymer melts in terms of complex fluidity

In dynamic rheological experiments melt behavior is usually expressed in terms of complex viscosity ^* () or complex modulusG ^* (). In contrast, we attempted to use the complex fluidity ^* () = 1/µ ^* () to represent this behavior. The main interest is to simplify the complex-plane diagram and to simplify the determination of fundamental parameters such as the Newtonian viscosity or the parameter of relaxation-time distribution when a Cole-Cole type distribution can be applied. ^* () complex shear viscosity - () real part of the complex viscosity - () imaginary part of the complex viscosity - G ^* () complex shear modulus - G() storage modulus in shear - G() loss modulus in shear - J ^* () complex shear compliance - J() storage compliance in shear - J() loss compliance in shear - shear strain - rate of strain - angular frequency (rad/s) - shear stress - loss angle - ^* () complex shear fluidity - () real part of the complex fluidity - () imaginary part of the complex fluidity - ₀ zero-viscosity - ₀ average relaxation time - h parameter of relaxation-time distribution     

Influence of vorticity on the heat transfer to blunt bodies in hypersonic flight

When blunt bodies are in hypersonic flight, a high-entropy layer of gas with nonzero vorticity is formed near their surface. The transverse gradients of the entropy, density, and gas velocity in the layer are high, which makes it necessary to take into account its absorption by the boundary layer of finite thickness . This vortex interaction is usually accompanied by an increase in the heat flux q and the frictional stress on the wall compared with their values as calculated in accordance with the classical scheme of a thin boundary layer, when the parameters on the outer edge of the boundary layer are set equal to the inviscid parameters on the body. This effect has been investigated on the side surface of slender (with angle 1 to the undisturbed flow) blunt bodies in a hypersonic stream [1–3]. It is shown in the present paper that the effect can have a stronger and even qualitative influence on the flow over blunt bodies with 1 if the radius of curvature R_s of the detached shock wave on the axis is small compared with the midsection radius R of the body. It is shown that the distributions of the heat fluxes with allowance for the vorticity of the inviscid shock layer are similar in the case of slightly blunt (r₀/R 0) cones with half-angles less than a critical _*.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 50–57, March–April, 1981.     

Effects of MHD free convection and mass transfer on the flow past an oscillating infinite coaxial vertical circular cylinder

The effects of MHD free convection and mass transfer are taken into account on the flow past oscillating infinite coaxial vertical circular cylinder. The analytical expressions for velocity, temperature and concentration of the fluid are obtained by using perturbation technique.

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Einwirkungen von freier MHD-Konvektion und Stoffübertragung auf eine Strömung nach einem schwingenden unendlichen koaxialen vertikalen Zylinder

Zusammenfassung Die Einwirkungen der freien MHD-Konvektion und Stoffübertragung auf eine Strömung nach einem schwingenden, unendlichen, koaxialen, vertikalen Zylinder wurden untersucht. Die analytischen Ausdrücke der Geschwindigkeit, Temperatur und Fluidkonzentration sind durch die Perturbationstechnik erhalten worden.

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Nomenclature C _p Specific heat at constant temperature - C the species concentration near the circular cylinder - C _w the species concentration of the circular cylinder - C the species concentration of the fluid at infinite - * dimensionless species concentration - D chemical molecular diffusivity - g acceleration due to gravity - Gr Grashof number - Gm modified Grashof number - K thermal conductivity - Pr Prandtl number - r _a ,r _b radius of inner and outer cylinder - a, b dimensionless inner and outer radius - r,r coordinate and dimensionless coordinate normal to the circular cylinder - Sc Schmidt number - t time - t dimensionless time - T temperature of the fluid near the circular cylinder - T _w temperature of the circular cylinder - T temperature of the fluid at infinite - u velocity of the fluid - u dimensionless velocity of the fluid - U ₀ reference velocity - z,z coordinate and dimensionless coordinate along the circular cylinder - coefficient of volume expansion - * coefficient of thermal expansion with concentration - dimensionless temperature - H ₀ magnetic field intensity - coefficient of viscosity - _e permeability (magnetic) - kinematic viscosity - electric conductivity - density - M Hartmann number - dimensionless skin-friction - frequency - dimensionless frequency     

Impacts of local dispersion and first-order decay on solute transport in randomly heterogeneous porous media

Stochastic subsurface transport theories either disregard local dispersion or take it to be constant. We offer an alternative Eulerian-Lagrangian formalism to account for both local dispersion and first-order mass removal (due to radioactive decay or biodegradation). It rests on a decomposition of the velocityv into a field-scale componentv , which is defined on the scale of measurement support, and a zero mean sub-field-scale componentv _s , which fluctuates randomly on scales smaller than. Without loss of generality, we work formally with unconditional statistics ofv _s and conditional statistics ofv . We then require that, within this (or other selected) working framework,v _s andv be mutually uncorrelated. This holds whenever the correlation scale ofv is large in comparison to that ofv _s . The formalism leads to an integro-differential equation for the conditional mean total concentration c which includes two dispersion terms, one field-scale and one sub-field-scale. It also leads to explicit expressions for conditional second moments of concentration cc. We solve the former, and evaluate the latter, for mildly fluctuatingv by means of an analytical-numerical method developed earlier by Zhang and Neuman. We present results in two-dimensional flow fields of unconditional (prior) mean uniformv . These show that the relative effect of local dispersion on first and second moments of concentration dies out locally as the corresponding dispersion tensor tends to zero. The effect also diminishes with time and source size. Our results thus do not support claims in the literature that local dispersion must always be accounted for, no matter how small it is. First-order decay reduces dispersion. This effect increases with time. However, these concentration moments c and cc of total concentrationc, which are associated with the scale below, cannot be used to estimate the field-scale concentrationc directly. To do so, a spatial average over the field measurement scale is needed. Nevertheless, our numerical results show that differences between the ensemble moments ofc and those ofc are negligible, especially for nonpoint sources, because the ensemble moments ofc are already smooth enough.     

Positively invariant regions for a problem in phase transitions

Positively invariant regions for the system v _t + p(W) _x = V _xx , W _t –V _x = W _xx are constructed where p < 0, w < , w > , p(w) = 0, w , > 0. Such a choice of p is motivated by the Maxwell construction for a van der Waals fluid. The method of an analysis is a modification of earlier ideas of Chueh, Conley, & Smoller [1]. The results given here provide independent L bounds on the solution (w, v).Dedicated to Professor James Serrin on the occasion of his sixtieth birthday     

Deformed particle image pattern matching in Particle Image Velocimetry

The deformation of particle image patterns due to velocity gradients causes errors of velocity measurements and false velocity detections in PIV (Particle Image Velocimetry). A novel technique to overcome those limitations inherent in the conventional PIV by correcting the particle image pattern according to the local velocity gradients in two dimensional flows, i.e. u/x, u/y, v/x and v/y, is proposed and successfully applied to a water flow downstream of a backward facing step.